Positive Energy Unitary Irreducible Representations Research Articles

Positive Energy Unitary Irreducible Representations of the Superalgebras osp(1|2n, IR) and Character Formulae for n = 3

We overview our study of the positive energy (lowest weight) unitary irreducible representations of the superalgebras osp(1|2n, IR). We give more explicitly character formulae for these representations in the case n = 3.

Positive energy unitary irreducible representations of the superalgebra osp(1|8,R)

We continue the study of positive energy (lowest weight) unitary irreducible representations of the superalgebras osp(1|2n,R). We present the full list of these UIRs. We give a proof of the case osp(1|8,R).

Ambitwistors, oscillators and massless fields on AdS5

Positive energy unitary irreducible representations of $SU(2,2)$ can be constructed with the aid of bosonic oscillators in (anti)fundamental representation of $SU(2)_L\times SU(2)_R$ that are closely related to Penrose twistors. Starting with the correspondence between the doubleton representations, homogeneous functions on projective twistor space and on-shell $SL(2,\mathbb C)$ generalized Weyl curvature spinors and their low-spin counterparts, we study in the similar way the correspondence between the massless representations, homogeneous functions on ambitwistor space and, via the Penrose transform, with the gauge fields on Minkowski boundary of $AdS_5$. The possibilities of reconstructing massless fields on $AdS_5$ and some applications are also discussed.

Explicit character formulae for positive energy unitary irreducible representations of D = 4 conformal supersymmetry

This paper continues the project of constructing the character formulae for the positive energy unitary irreducible representations of the N-extended D = 4 conformal superalgebras su(2, 2/N). In the first paper we gave the bare characters which represent the defining odd entries of the characters. Now we give the full explicit character formulae for N = 1 and for several important examples for N = 2 and N = 4.

Group-theoretical classification of BPS and possibly protected states in [formula omitted] conformal supersymmetry

We give explicitly the reduction of supersymmetries of the positive energy unitary irreducible representations of the N-extended D = 4 conformal superalgebras su ( 2 , 2 / N ) . Further we give the classification of BPS and possibly protected states.

Minimal unitary representation of SU(2,2) and its deformations as massless conformal fields and their supersymmetric extensions

We study the minimal unitary representation (minrep) of SO(4,2) over a Hilbert space of functions of three variables, obtained by quantizing its quasiconformal action on a five dimensional space. The minrep of SO(4,2), which coincides with the minrep of SU(2,2) similarly constructed, corresponds to a massless conformal scalar in four space-time dimensions. There exists a one-parameter family of deformations of the minrep of SU(2,2). For positive (negative) integer values of the deformation parameter ζ, one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in (0,ζ∕2)((−ζ∕2,0)) representation of the SL(2,C) subgroup. We construct the supersymmetric extensions of the minrep of SU(2,2) and its deformations to those of SU(2,2|N). The minimal unitary supermultiplet of SU(2,2|4), in the undeformed case, simply corresponds to the massless N=4 Yang–Mills supermultiplet in four dimensions. For each given nonzero integer value of ζ, one obtains a unique supermultiplet of massless conformal fields of higher spin. For SU(2,2|4), these supermultiplets are simply the doubleton supermultiplets studied in the work of Gunaydin et al. [Nucl. Phys. B 534, 96 (1998); e-print arXiv:hep-th/9806042].

Minimal unitary representation of [formula omitted] and its [formula omitted] deformations as massless 6 D conformal fields and their supersymmetric extensions

We study the minimal unitary representation (minrep) of SO ( 6 , 2 ) over an Hilbert space of functions of five variables, obtained by quantizing its quasiconformal realization. The minrep of SO ( 6 , 2 ) , which coincides with the minrep of SO ∗ ( 8 ) similarly constructed, corresponds to a massless conformal scalar field in six spacetime dimensions. There exists a family of “deformations” of the minrep of SO ∗ ( 8 ) labeled by the spin t of an SU ( 2 ) T subgroup of the little group SO ( 4 ) of lightlike vectors. These deformations labeled by t are positive energy unitary irreducible representations of SO ∗ ( 8 ) that describe massless conformal fields in six dimensions. The SU ( 2 ) T spin t is the six-dimensional counterpart of U ( 1 ) deformations of the minrep of 4D conformal group SU ( 2 , 2 ) labeled by helicity. We also construct the supersymmetric extensions of the minimal unitary representation of SO ∗ ( 8 ) to minimal unitary representations of OSp ( 8 ∗ | 2 N ) that describe massless six-dimensional conformal supermultiplets. The minimal unitary supermultiplet of OSp ( 8 ∗ | 4 ) is the massless supermultiplet of ( 2 , 0 ) conformal field theory that is believed to be dual to M-theory on AdS 7 × S 4 .

Characters of the positive energy UIRs of D = 4 conformal supersymmetry

We give character formulas for the positive energy unitary irreducible representations of the N-extended D = 4 conformal superalgebras su(2, 2/N). Using these, we also derive decompositions of long superfields as they descend to the unitarity threshold. These results are also applicable to irreps of the complex Lie superalgebras sl(4/N). Our derivations use results from the representation theory of su(2, 2/N) developed as early as the 1980s.

Decompositions of superfields inD=4 conformal SUSY

We use character formulae for the positive-energy unitary irreducible representations of theN-extendedD=4 conformal superalgebras su(2, 2/N) that were derived earlier to give decompositions of long superfields as they descend to the unitarity threshold.

Positive energy unitary irreducible representations of the superalgebras osp(1|,2n, ℝ)

We give the classification of the positive energy (lowest weight) unitary irreducible representations of the superalgebras osp(1|2n,R).

Could only fermions be elementary?

In standard Poincare and anti de Sitter SO(2,3) invariant theories, antiparticles are related to negative energy solutions of covariant equations while independent positive energy unitary irreducible representations (UIRs) of the symmetry group are used for describing both a particle and its antiparticle. Such an approach cannot be applied in de Sitter SO(1,4) invariant theory. We argue that it would be more natural to require that (*) one UIR should describe a particle and its antiparticle simultaneously. This would automatically explain the existence of antiparticles and show that a particle and its antiparticle are different states of the same object. If (*) is adopted then among the above groups only the SO(1,4) one can be a candidate for constructing elementary particle theory. It is shown that UIRs of the SO(1,4) group can be interpreted in the framework of (*) and cannot be interpreted in the standard way. By quantizing such UIRs and requiring that the energy should be positive in the Poincare approximation, we conclude that i) elementary particles can be only fermions. It is also shown that ii) C invariance is not exact even in the free massive theory and iii) elementary particles cannot be neutral. This gives a natural explanation of the fact that all observed neutral states are bosons.

Positive energy unitary irreducible representations of D $equal$ 6 conformal supersymmetry

We give a constructive classification of the positive energy (lowest weight) unitary irreducible representations of the D=6 superconformal algebras osp(8*/2N). Our results confirm all but one of the conjectures of Minwalla (for N=1,2) on this classification. Our main tool is the explicit construction of the norms of the states that has to be checked for positivity. We give also the reduction of the exceptional UIRs.

Unitary supermultiplets of [formula omitted] and the spectrum of the S 7 compactification of 11-dimensional supergravity

The spectrum of the S 7 compactification of 11-dimensional supergravity is constructed very simply by using the oscillator method. The spectrum consists of a special class of positive energy unitary irreducible representations (UIR's) of the N = 8 anti-de Sitter supergroup OSp( 8 4 , R) . This class of UIR's correspond to the short supermultiplets whose spin range is 2. The ultra-short supermultiplet, the singleton, whose spin range is 1 2 has no Poincaré limit and does not occur in the spectrum. The method described here can be used to obtain all positive energy unitary supermultiplets of OSp( 8 4 , R) and not just those that arise in the S 7 compactification. In addition, the oscillator method can be used to obtain the positive energy UIR's of the non-compact supergroups OSp( 2n 2m , R) , OSp( 2n ∗ 2m ) and SU(n, m p ) .

All positive energy unitary irreducible representations of extended conformal supersymmetry

We give the list of all positive energy UIR's of the conformal superalgebra su(2, 2 N ) . They are realized as subrepresentations of elementary representations. The latter are induced from irreducible finite-dimensional Lorentz and su( N) representations on spaces of functions (superfields). Some of the unitary representation spaces are comprised from solutions of invariant differential equations. Among these the massless UIR's are discussed in detail.